Distribution Frames and Bases

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2 Citazioni (Scopus)

Abstract

In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate conditions for them to constitute a ”continuous basis” for the smallest space D of a rigged Hilbert space. This analysis requires suitable extensions of familiar notions as those of frames, Riesz bases and orthonormal bases. A motivation for this study comes from the Gel’fand–Maurin theorem which states, under certain conditions, the existence of a family of generalized eigenvectors of an essentially self-adjoint operator on a domain D which acts like an orthonormal basis of the Hilbert space H. The corresponding object will be called here a Gel’fand distribution basis. The main results are obtained in terms of properties of a conveniently defined synthesis operator
Lingua originaleEnglish
pagine (da-a)2109-2140
Numero di pagine32
RivistaJournal of Fourier Analysis and Applications
Volume25
Stato di pubblicazionePublished - 2019

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Hilbert spaces
Orthonormal basis
Hilbert space
Riesz Basis
D-space
Self-adjoint Operator
Eigenvalues and eigenfunctions
Eigenvector
Synthesis
Operator
Theorem

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics(all)
  • Applied Mathematics

Cita questo

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title = "Distribution Frames and Bases",
abstract = "In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate conditions for them to constitute a ”continuous basis” for the smallest space D of a rigged Hilbert space. This analysis requires suitable extensions of familiar notions as those of frames, Riesz bases and orthonormal bases. A motivation for this study comes from the Gel’fand–Maurin theorem which states, under certain conditions, the existence of a family of generalized eigenvectors of an essentially self-adjoint operator on a domain D which acts like an orthonormal basis of the Hilbert space H. The corresponding object will be called here a Gel’fand distribution basis. The main results are obtained in terms of properties of a conveniently defined synthesis operator",
author = "Salvatore Triolo and Camillo Trapani and Francesco Tschinke",
year = "2019",
language = "English",
volume = "25",
pages = "2109--2140",
journal = "Journal of Fourier Analysis and Applications",
issn = "1069-5869",
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T1 - Distribution Frames and Bases

AU - Triolo, Salvatore

AU - Trapani, Camillo

AU - Tschinke, Francesco

PY - 2019

Y1 - 2019

N2 - In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate conditions for them to constitute a ”continuous basis” for the smallest space D of a rigged Hilbert space. This analysis requires suitable extensions of familiar notions as those of frames, Riesz bases and orthonormal bases. A motivation for this study comes from the Gel’fand–Maurin theorem which states, under certain conditions, the existence of a family of generalized eigenvectors of an essentially self-adjoint operator on a domain D which acts like an orthonormal basis of the Hilbert space H. The corresponding object will be called here a Gel’fand distribution basis. The main results are obtained in terms of properties of a conveniently defined synthesis operator

AB - In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate conditions for them to constitute a ”continuous basis” for the smallest space D of a rigged Hilbert space. This analysis requires suitable extensions of familiar notions as those of frames, Riesz bases and orthonormal bases. A motivation for this study comes from the Gel’fand–Maurin theorem which states, under certain conditions, the existence of a family of generalized eigenvectors of an essentially self-adjoint operator on a domain D which acts like an orthonormal basis of the Hilbert space H. The corresponding object will be called here a Gel’fand distribution basis. The main results are obtained in terms of properties of a conveniently defined synthesis operator

UR - http://hdl.handle.net/10447/371264

UR - https://link.springer.com/article/10.1007/s00041-018-09659-5

M3 - Article

VL - 25

SP - 2109

EP - 2140

JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

SN - 1069-5869

ER -